| L(s) = 1 | + i·3-s + 5-s + (1 + 2.44i)7-s − 9-s − 2·11-s + 6.89·13-s + i·15-s − 4.89i·17-s − 6.89i·19-s + (−2.44 + i)21-s − 6i·23-s + 25-s − i·27-s − 8.89i·29-s − 2·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447·5-s + (0.377 + 0.925i)7-s − 0.333·9-s − 0.603·11-s + 1.91·13-s + 0.258i·15-s − 1.18i·17-s − 1.58i·19-s + (−0.534 + 0.218i)21-s − 1.25i·23-s + 0.200·25-s − 0.192i·27-s − 1.65i·29-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.046695596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.046695596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 6.89iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 8.89iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 7.79iT - 37T^{2} \) |
| 41 | \( 1 + 2.89iT - 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 9.79iT - 53T^{2} \) |
| 59 | \( 1 - 3.10iT - 59T^{2} \) |
| 61 | \( 1 + 0.898T + 61T^{2} \) |
| 67 | \( 1 - 4.89T + 67T^{2} \) |
| 71 | \( 1 - 12.8iT - 71T^{2} \) |
| 73 | \( 1 - 8.89iT - 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 1.10iT - 89T^{2} \) |
| 97 | \( 1 - 0.898iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.667261700917137734692002360832, −8.103284631705298564429962346739, −6.98174725827152296465513354929, −6.15061982122500072063404334700, −5.51852055777637463575538943162, −4.82818498611878239081539190874, −3.96459563471962245034649395768, −2.80153395349673607882482377546, −2.24078114780081284012721942793, −0.64776795670793888190028680830,
1.41013107511985705699499992600, 1.55490807684880440457249402100, 3.29404865414676675108920186287, 3.76682542561194716928389880873, 4.93038117692219436572411995923, 5.91609918605333307498718579149, 6.25075159919035403825449611042, 7.25403427035311819748239730550, 8.029309970149310554250646057994, 8.376875482300715591632106126138
